Asymptotic model for the interplay of sedimentation and crystallization

This paper considers a simple model for the coupled crystallization and sedimentation of a nucleated particle in a solvent. It is motivated by recent interest in understanding the effects of gravity on crystallization, arising in the context of possible future extraterrestrial manufacture of active pharmaceutical ingredients. The model is based on massand momentum-conservation arguments and consists of a nonlinear ordinary differential equation and an integrodifferential equation for the particle radius and its velocity, respectively, as functions of time; the second of these equations contains the well-known Basset integral history term. Nondimensionalization and subsequent asymptotic analysis indicate that there can be up to four different regimes as the growing particle sediments. The results of the analysis are compared with numerical solutions and found to give very good agreement; this is done for the crystallization of L-histidine, an essential amino acid that is used in the food and pharmaceutical industries. The availability of asymptotic solutions is particularly attractive in view of the surprisingly lengthy computational time, which is directly associated with the cumulative computational burden of the history term. An important implication is that the forced convection considered here outweighs any free convection effects arising from solute concentration gradients. Moreover, depending on the strength of the gravitational field, e.g., hypergravity or microgravity, the particle can either sediment, then grow, or grow while sedimenting. This distinction is critical for future models, as determining whether the two processes occur concurrently or sequentially influences the mathematical description of the crystallization dynamics.